Alessandro Oneto: Combinatorial tools for new questions on planar polynomial interpolation

Polynomial interpolation problems have been largely studied in algebraic geometry and commutative algebra. The classical question is the following:

How many conditions does a general union of fat points give on the complete linear system of plane curves of given degree?

The answer to this question is given by the famous SHGH Conjecture which has been proven to be true in some cases but, in general, is still open. In a recent paper, D. Cook II, B. Harbourne, J. Migliore and U. Nagel, started to investigate a different question by looking at the conditions imposed by a general fat point to the incomplete linear system of curves of given degree passing through a given set of points X (not in general position). There are cases in which the expected number of conditions is not achieved and we have unexpected curves. In their work, they relate the existence of unexpected curves with properties of the line arrangement dual to the given set of points X. In particular, to the exponents, or splitting type, of the arrangement.

In this talk, after describing the problem and the relation between unexpected curves and line arrangements, I will present a joint ongoing project with Michela Di Marca (U. of Genoa, Italy) and Grzegorz Malara (Pedagogical U. of Cracow, Poland). We classify supersolvable line arrangements whose dual configuration of points admits unexpected curves and we provide new families of line arrangements having this unexpected property.